2010/10/31 by Gérard Ben Arous, Paul Bourgade
Mathematics · Physics and Astronomy · #Analytic Number Theory Research #Circular ensemble #Constant (computer programming) #Dimension (graph theory) #Eigenvalues and eigenvectors #Gaussian #Limiting #Matrix (chemical analysis) #Random Matrices and Applications #Random matrix #Stochastic processes and statistical mechanics #Unitary matrix #Unitary state #math-ph #math.MP #math.PR
paper · pdf · doi:10.1214/11-aop710
published as Annals of Probability 2013, Vol. 41, No. 4, 2648-2681 · Published in at http://dx.doi.org/10.1214/11-AOP710 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)
openalex publication_date 2013/07/01 · arxiv created 2013/07/24 · arxiv updated 2013/07/25 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/06
This paper studies the extreme gaps between eigenvalues of random matrices. We give the joint limiting law of the smallest gaps for Haar-distributed unitary matrices and matrices from the Gaussian unitary ensemble. In particular, the kth smallest gap, normalized by a factor n-4/3, has a limiting density proportional to x3k-1e^-x3. Concerning the largest gaps, normalized by n/√(log n), they converge in L p to a constant for all p>0. These results are compared with the extreme gaps between zeros of the Riemann zeta function.